Labour revenue spokesman Clayton Cosgrove said Mr Dunne could not escape responsibility just because he lost the revenue portfolio when he resigned temporarily from Prime Minister John Key's ministry in June 2013 over an allegation that he leaked a security report to a journalist.
He is now Minister of Internal Affairs and National MP Todd McClay is Revenue Minister.
"It's very interesting that Peter Dunne held this portfolio for many, many years, and as soon as there is a change of minister we now find these things out," Mr Cosgrove said.
He said Mr McClay should now explain how the cost blowout occurred and answer whether the whole reform was now justified.
He also questioned whether the Inland Revenue Department was using its allegedly "clunky" old computer system as "an alibi" to escape responsibility for the cost blowout.
The department wants to spend $1.5 billion on a "business transformation" that will include a completely new computer system, and the regulatory impact statement issued with last week's child support amending bill said that on top of the $163 million 10-year cost of the child support reforms, "there will be an additional cost of migrating the reforms to the new 'transformed environment'."
"What has become an alibi for doing nothing, and for cost blowouts, is that they are going to dispose of the old computer system," Mr Cosgrove said.
"I don't think there is one bank that has disposed of its legacy IT system. They tend to make incremental add-ons to it."
However Mr Dunne said he did not know how the cost of implementing the child support reform had blown out to $163 million.
"At the time I left in June 2013 we had agreed to a figure, which had been through Cabinet, that the scheme changes would cost $120 million, spread pretty evenly over a 10-year period," he said.
"How that figure became $163 million, and the change in the balance of when it is being spent, I simply don't have that detail."
He said Cabinet "was prepared to wear" the initial cost increase from $30 million to $120 million, which reflected "the complexity of the changes".
"Every transaction done under the system as it stands today per client is one transaction," he said.
"As I understand it, under the new system that one transaction multiplies to seven or eight. You work that out over 10 years and you can see why it costs more.
"The current system says, 'Eligible? Yes or no,' basically. This [reform] takes more account of income being earned by custodial parents as well as non-custodial parents. There is a whole range of other issues there to be checked through before reaching a decision."
He said that even at $163 million, the reform was still worth doing because it would be fairer to all parents and their 200,000 children.
